Questions on the mathematics behind the famed toy invented by Ernő Rubik.

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How many ways are there to solve a Rubiks cube?

Friends have been arguing that a rubiks cube could never be mastered cause there are too many different ways to solve it cause there are 43,252,003,274,489,856,000 positions the Cube could have, so ...
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1answer
72 views

How many combinations in 10x10x10 Rubik's cube?

I was wondering how many possible combinations there is in the cubes greater than 3x3x3 (4x4x4, 5x5x5, ..., 10x10x10)? We know that in 3x3x3 there are about 4,3 * 10^19 combinations, what about bigger ...
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2answers
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Rubik's Cube's Group

Is there an article somewhere with an exhaustive study of the Rubik's Cube Group $G$? Such as computing some subgroups of it or exhibiting some elements of its center $Z(G)$? I tried googling it and ...
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1answer
53 views

Rubik's Cube Thistlethwaite Four Phase Algorithm

I am trying to understand The Four-Phase-Algorithm ( Thistlethwaite's 52-move algorithm ) for Rubik's Cube but no easy understandable explaination is given. It would be helpful if given with a small ...
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how to evaluate combinations of different configurarations of a rubik's cube

how to calculate the total combination number of an rubik's cube ? and what is the number of combination if we fix one face of it with all matched colours , specifically saying blue one ,?? no. of ...
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1answer
115 views

$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?

I've been working on a problem related to the 3x3x3 Rubik's Cube where you allow faces to be turned by $45^\circ$ instead of just the usual $90^\circ$. We know for the standard 3x3x3 the cube is ...
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32 views

Can solution to Rubik's cube be seen from the point of view of Markov Decision Process?

Solving Rubik's cube can be thought of as a Planning problem which has : a state space $S$ a set $G \subseteq S$ of goal states (in this case singleton) actions $A(s) \subseteq A$ applicable in ...
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1answer
32 views

Imitating smaller Rubik's cubes with bigger ones.

Let's assume we have a 4x4x4 Rubik's cube. With this cube we can imitate a 2x2x2 by considering all 2x2x2 corner cubes as 1 block. Similarly, we can solve it like a 3x3x3 by first solving the 2x2 ...
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1answer
101 views

How to learn about the Rubik's Cube?

So I just started learning Abstract Algebra and I've recently appreciated an introduction to group theory. Among the interesting avenues of group theory is found in tackling a Rubik's Cube. I've ...
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35 views

Help with proving property of Rubik's cube.

I am looking at these notes, and am trying to prove the following on page 13: Let $C_1$ and $C_2$ be two different unoriented corner cubies, and let $C'_{1}$ and $C'_{2}$ be two different ...
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1answer
129 views

Solve All Sequence (Rubik's Cube)

Can you prove/disprove that there is a solve-all sequence of moves to complete the Rubik's Cube from any solvable-position? If so, can you explain how long it is? If not, explain why not. Just to be ...
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2answers
153 views

Order of the Rubik's cube group

Associated to the Rubik's cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise ...
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1answer
141 views

Where Does F' in Rubik's Cube Group Singmaster Notation Come From?

Basic 90° : F turns the front clockwise 180° : F^2 turns the front clockwise twice -90° : F' turns the front counter-clockwise Why are we calling the -90° rotation F' and not -F? (source: ...
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0answers
64 views

Rubik's cube computing algorithm

Is it possible to solve a 3x3 Rubik's cube using a tree similar to a 'mini-max game tree' (listing all possible unique moves, then listing further moves from this and so on), or is the sample space ...
2
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0answers
127 views

Minimum number of stickers required for 3x3 Rubik's Cube

Lately, the stickers on my V-Cube have been peeling off, and I became curious: what is the minimum number of stickers I would actually need in order to represent each unique configuration? I have but ...
6
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1answer
361 views

Miniature Rubik's Cube

$R$ is a clock-wise twist of a right-hand face and $U$ is a clock-wise twist of the upper face of a $2 \times 2 \times 2$ Rubik's Cube. Find the order of the subgroup generated by $\{R, U\}$
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1answer
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How to use group theory to solve larrys square iphone app

There's a 2 d version of rubiks cube on apple app store. How can group theory give an algorithm to solve the iPhone app:larry's square.
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2answers
188 views

Is there a proof, that for any Rubik's cube, after solving the first 2 layers, there will always be an even amount of oriented edges?

I've always wondered if there was a proof for this, since the beginners method to solving the last layer depends on this (you either have no edges oriented correctly, an L, a line or a cross, and each ...
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1answer
207 views

Complex automata Rubiks cube question (with picture) help needed

Question 1 A Rubik’s Cube is a puzzle in the shape of a cube. Each face is covered by nine stickers, each of which is coloured with one of six colours: white, red, blue, orange, green, andyellow. An ...
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1answer
231 views

Number of unique permutations of a 3x3x3 cube

Given a 3x3x3 cube (like a rubik's cube) where each of the 27 cubes has a distinct number, how many unique permutations are possible? Simple rotations of the entire cube should not be counted. The ...
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identifying a subgroup of $S_8$ generated by 4-cycles

Let $G \subseteq S_8$ be the subgroup generated by some 4-cycles. If we number the elements $1,2,\dots, 8$, the 4-cycles are $(1234),(5678),(1485),(2376),(1265),(4378)$ I am not sure if I have ...
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1answer
1k views

Creating a Rubik's Cube Algorithm

So, I realize this isn't a very math-oriented question, but I thought it fit better here than anywhere else. Anyway, I was simply wondering how people come up with the many Rubik's Cube algorithms, ...
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Rubik's cube with different neighboring color

How many valid configuration of the 3x3 Rubik's cube are there so that no two adjacent faces are the same color? Thanks.
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3answers
634 views

Minimum number of random moves needed to uniformly scramble a Rubik's cube?

Follow up from my last question: $3 \times 3$ Rubik's cube scrambling question I am talking about $3 \times 3$ Rubik's cubes. Start with a solved cube. Then make some amount of random moves ...
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$3 \times 3$ Rubik's cube scrambling question

It was proven that a Rubik's cube needs at most $20$ moves to solve. This implies that any configuration of a Rubik's cube can be reached from an unscrambled Cube in at most $20$ moves. So, say when ...
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1answer
118 views

Eulerian path for Rubik's Cube states

There are a number of discussions online confirming that there exists a Hamiltonian cycle through the states of a Rubik's Cube. Or more precisely, the "quarter-turn metric Cayley graph for the Rubik's ...
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2answers
930 views

Can RUBIK's cube be solved using group theory?

Can RUBIK's cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?
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1answer
67 views

Independently rotating cubelets of a 2 x 2 rubiks cube?

I was reading up on 2 x 2 Rubik's cube. Wikipedia says that Any permutation of the eight corners is possible (8! positions), and seven of them can be independently rotated (37 positions). I ...
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1answer
83 views

What is “tetrad twist” in relation to Rubik's cubes?

I am trying to write a computer program to solve Rubik's cubes using a version of Thistlethwaite's algorithm. Instead of using lookup tables to perform each phase of the algorithm, I'm using a ...
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3answers
3k views

Can a Rubik's cube be mapped knowing only two sides?

Is it possible to know the entire configuration of a Rubik's cube looking at only two sides and not rotating the cube? In other words: what is the minimum information required to create a ...
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1answer
154 views

What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;] Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn) ...
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91 views

Rubik cube number of alternative solutions

If a cube is in a configuration that requires 20 moves to solve, is that sequence unique, or are there multiple sequences that arrive at a solution? That is: are there are two or more sequences that ...
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2answers
814 views

Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?

I'm not sure if this question belongs here but I see lots of Rubik Cube's questions around so here it goes: Can I take a standard 3x3 Rubik's Cube and shuffle it so that, for every face, there are no ...
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4answers
312 views

Unreachable rubik cube positions.

Are there positions of the rubik cube which cannot be reached by applying the standard moves starting from the solved cube? If so, how many such positions are there?
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1answer
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Rubik's cube question showing surjectivity

Suppose $f: G \rightarrow S_C$ be the action of Rubik's group G on the collection $C$ of corner cubelets. I want to show that the action $f$ is onto $S_C$.
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1answer
131 views

Rubik's Group Question - Commutators

Suppose that $f: G \rightarrow S_C$ be the action of Rubik's group $G$ on the collection of $C$ of corner cubelets. So if $C_1$ and $C_2$ be distinct corner cubelets. Show that there is a move $M \in ...
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2answers
190 views

What are the total number of distinguishable configurations of a rubik's cube?

Given a standard $3 \times 3 \times 3$ rubik's cube, what are the total number of distinguishable configurations of the cube? Two configurations are called distinguishable iff one can be rotated to ...
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0answers
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How to “validate” a Rubik's Cube configuration? [duplicate]

Looking at a picture of a Rubik's cube how can I tell if it's in one of its possible configurations (image is not fake)? By "possible configuration" I mean a configuration that can be achieved by ...
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0answers
157 views

Faces on a rotating cube

This one is probably a softball question, but... Fact: A cube has 6 faces Fact (unless my math is wrong): A cube that is cut in half center horizontal can be reoriented to have 18 unique faces. ...
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2answers
544 views

combinations of Rubik cube

A Rubick's Cube has owl heads on it, which can be misoriented. How many (times) MORE combinations are there of this cube vs. one that has blank stickers? Can someone give me some hints? Thanks
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1answer
368 views

Rubik's cube and counting

Inspired by this question, I formulate the following: Suppose I have a $3\times3\times3$ Rubik's cube, call each small square on the surface a piece, there are then $3*3*6 = 54$ pieces. Enumerate ...
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1answer
601 views

Literature on group theory of Rubik's Cube

While searching for literature on the group theory of Rubik's Cube, I mostly find introductions to group theory motivated by applications to Rubik's cube. I.e. the focus lies on elementary group ...
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Why should Rubik's cube get attention from mathematicians?

I've seen a lot of math debate, calculations and other stuff related to Rubik's cube lately, but I don't really understand why is it important, why should anyone spend time asking and answering ...
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870 views

Rubik Cube finite non-abelian group

I was reading the following paper from MIT: http://web.mit.edu/sp.268/www/rubik.pdf The paper is not difficult to understand, it is more or less a short introduction into group theory, taking the ...
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1answer
172 views

Rubik's cube as $2\times2\times2$ or $2\times2$? [closed]

I'm surprised why people call a Rubik's cube as $2 \times2\times2$ or $3\times3\times3$? It's a CUBE! Isn't saying $2\times2$ or $3\times3$ cube (or rather just $2$ or $3$) sufficient?
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2answers
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Lower bounds on numbers of arrangements of a Rubik's cube

Last night, a friend of mine informed me that there were forty-three quintillion positions that a Rubik's Cube could be in and asked me how many there were for my Professor's Cube (5x5x5). So I gave ...
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4answers
3k views

Minimum number of steps needed to solve a rubik cube

Long time ago I've seen a book on group theory and there was an appendix about rubik cube. I remember there were only three steps that enabled me to solve my cube (three strings with letters encoding ...
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1answer
157 views

The redundancy of Rubik's cube states [duplicate]

Possible Duplicate: Rubik’s Cube Not a Group? I take a Rubik's cube in the solved state, and I secretly assign a unique integer label to each of the cubies. I then, via an arbitrarily long ...
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2answers
616 views

Rubik's Cube Combination

Could anyone explain why the number of legal or reachable combinations of a $3\times 3\times 3$ Rubik's Cube is $1/12\mbox{th}$ of the total. I understood the logic behind the total number of ...
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Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...